Note that \(\mbox{cx}\,_R(M)=\mbox{cx}\,_R(\Omega ^i_R(M))\)  for every \(i\ge 0\) . It follows from the definition that
\(\mbox{cx}\,_R(M)=0\)  if and only if \(\mbox{Pd}\,_R(M)<\infty \) .
The complete intersection dimension was introduced by Avramov, Gasharov and Peeva [1]}. A module of finite complete intersection dimension behaves homologically like a module over a complete intersection.
Recall that a quasi-deformation of \(R\)  is a diagram \(R\rightarrow A\twoheadleftarrow Q\)  of local homomorphisms, in which
\(R\rightarrow A\)  is faithfully flat, and \(A\twoheadleftarrow Q\)  is surjective with kernel generated by a regular sequence.
The module \(M\)  has finite complete intersection dimension if there exists such a quasi-deformation for which \(\mbox{Pd}\,_Q(M\otimes _RA)\)  is finite.
The complete intersection dimension of \(M\) , denoted \(\mbox{CI-dim}\,_R(M)\) , is defined as follows;
\(\mbox{CI-dim}\,_R(M)=\inf \lbrace \mbox{Pd}\,_Q(M\otimes _RA)-\mbox{Pd}\,_Q(A)\mid R\rightarrow A\twoheadleftarrow Q \text{ is a quasi-deformation }\rbrace .\) 
